AN  INVESTIGATION 


OF  THE 

ONE-HINGED  STEEL  ARCH 

AND  ITS 

COMPARISON  WITH  OTHER  TYPES 


A THESIS 


presented  to  the 

Faculty  of  the  Graduate  School 
Cornell  University 

for  the  degree  of 

Dodtor  of  Philosophy 

by 

Nee  Sun  Koo,  B.S.,  M.C.E., 


McGraw  Fellow  in  Civil  Engineering,  1920-’2 1 
1921 


Reprinted  from  the  Cornell  Civil  Engineer, 
volume  29,  pages  110,  129,  150,  March,  April,  May,  1921 


Digitized  by  the  Internet  Archive 
in  2017  with  funding  from 

University  of  Illinois  Urbana-Champaign  Alternates 


https://archive.org/details/investigationofoOOkoon 


AN  INVESTIGATION  OF  THE  ONE-HINGED  ARCH  AND 
ITS  COMPARISON  WITH  OTHER  TYPES 

By  Nee  Sun  Koo,  B.  S.,  M.  C.  E.  (1919). 

McGraw  Fellow  in  Civil  Engineering,  1920-21. 

An  Abstract  of  a Thesis  to  be  Presented  to  the  Faculty  of  the  Graduate  School  of  Cornell  University  for 

the  Degree  of  Doctor  of  Philosophy. 


PREFACE 

During  the  last  thirty  years  much  study  has  been 
given  to  the  design  and  construction  of  steel  arches 
by  American  engineers  and  investigators.  Two- 
hinged  and  three-hinged  arches  seem  to  have  met 
the  greatest  favor,  while  few  no-hinged  steel  arches 
have  been  built  in  this  country.  One-hinged  arches 
are  practically  unknown  in  America,  although  a few 
have  been  successfully  constructed  on  the  European 
continent.  After  considerable  study,  the  author 
has  been  in  doubt  of  the  practicability  and  value 
of  the  one-hinged  steel  arch.  He  could  see  no 
logical  or  conspicuous  reason  against  its  adoption. 
A number  of  writers  attack  it  bitterly  while  enumer- 
ating a few  disadvantages,  but  fail  to  demonstrate 
or  justify  their  statements.  Others  deem  it  un- 
necessary to  give  a full  treatment,  because  it'  has 
not  come  into  popular  use.  The  deficiency  of 
theoretical  knowledge  may  be  the  reason  why  the 
building  of  one-hinged  steel  arches  has  been  at- 
tempted so  rarely.  With  this  idea  in  mind,  the 
author  has  undertaken  a special  investigation  of 
the  one-hinged  steel  arch. 

While  this  work  is  undertaken  purely  for  the 
purpose  of  discovering  and  publishing  some  un- 
known facts,  some  contributions  made  by  previous 
investigators  will  be  mentioned.  One  of  the  noblest 
and  most  remarkable  preliminary  designs  which 
have  ever  been  made  in  bridge  engineering  was  con- 
tributed by  Charles  Worthington  in  competition 
with  other  designs  for  the  famous  Quebec  Bridge. 
It  was  a one-hinged  steel  arch  with  a span  of  1,800 
feet,  more  than  .twice  as  long  as  that  of  any  arch 
bridge  then  existing  in  the  world.  One  of  the  most 
noted  American  bridge  engineers.  Dr.  B.  A.  L.  Wad- 
dell, in  his  book  called  “Bridge  Engineering,” 
praises  the  work  of  Mr.  Worthington  and  calls  the 
design  ingenious  and  quite  feasible.  It  is  to  be 
regretted  that  the  scheme  was  not  accepted  by  the 
Canadian  Government  and  thus  the  complete  treat- 
ment of  the  theoretical  and  practical  problems  in- 
volved was  prevented.  The  work  gives  some  indi- 
cation of  what  could  be  done  with  the  one-hinged 
arch  and  done  economically.  Dr.  Waddell’s  com- 
ment is  valuable  on  account  of  his  extensive  ex- 
perience and  gives  promise  for  the  future.  In  view 
of  these  facts,  the  author  felt  encouraged  to  carry 
on  this  investigation. 

Special  emphasis  is  laid  upon  two  points,  first, 


to  make  an  extensive  study  of  its  behavior  in  carry- 
ing the  load ; and  second,  to  reveal  its  characteristics 
by  critical  comparisons  with  other  types  of  arches. 
It  is  his  hope  that  the  work  may  be  of  value  to  the 
engineering  profession  in  the  future,  if  not  at  pres- 
ent, Since  so  little  has  been  written  in  engineer- 
ing literature  upon  the  subject,  it  is  hoped  that 
every  bit  of  the  original  work  in  this  thesis  will 
appeal  to  the  sympathetic  interest  of  engineers  and 
investigators.  Acknowledgment  is  due  to  Profes- 
sor Henry  S.  Jacoby,  chairman  of  the  special  com- 
mittee in  charge  of  the  author’s  graduate  work  at 
Cornell  University,  for  his  aid  and  helpful  sugges- 
tions. 

Reactions  Solved  by  the  Author’s  Method  of 
Symmetrical  Deflection  Equations. 

Two  kinds  of  beams,  either  straight  or  curved, 
are  used  in  bridge  building;  those  which  are 
statically  determinate,  and  those  statically  inde- 
terminate. For  the  solution  of  beams  of  the  first 
class,  three  statical  equations  are  available: — 


= O (a> 

EH-  O (b) 


EM  O 


(c) 


For  the  solution  of  beams  of  the  second  class,  three 
elastic  conditions  are  available,  besides  the  statical 
conditions.  These  are 


Ay 


-f- 

f 


Pixels 


El 

Plds 


(d) 

(e) 

(f) 


where  Ay  is  the  horizontal  deflection,  Ay  the  ver- 
tical deflection,  and  jo  the  change  of  the  slope  be- 
tween any  two  points  on  the  beam,  E the  modulus 
of  elasticity  of  its  material,  I the  moment  of  in- 
ertia of  a section,  and  .1/  the  moment  at  any  point 
between  a and  b , where  I is  taken.  By  means  of 
these  six  fundamental  equations,  all  beams  can  be 
solved.  (For  the  derivation  of  formulas  (d),  (e) 
and  (/)  see  standard  books  on'  Mechanics).  The 


author  lias  used  these  equations  in  deriving  general 
formulas  of  reactions  for  no-,  one-,  two-,  and  three- 
hinged  arches. 

Let  an  arch-rib  with  a span  l and  rise  h be  fixed 
at  two  ends  a and  b,  and  hinged  at  the  crown  c and 
be  subject  to  a vertical  load  P at  a distance  kl  from 
the  left  end.  (Fig.  1).  The  load  is  sustained  by 
the  reactions  Hx,  Vx  and  Mx  at  the  left  support,  and 
by  H2,  V and  M2  at  the  right  support.  There  are 
six  unknowns  and  six  conditions  are  required  for 
its  solution.  The  three  statical  equations  furnish 
three  conditions,  while  the  hinge  at  the  crown  in- 
sures that  the  moment  at  that  point  is  zero.  It 
remains  for  us  to  find  two  more  elastic  conditions. 
In  other  words,  we  must  choose  any  two  equations 
from  ( d ),  (e)  and  (/)  and  apply  them  to  the  case 
of  the  one-hinged  arch. 

In  order  to  simplify  the  algebraic  work,  the  fol- 
lowing notations  are  proposed  by  the  author: — 


vertical  line  at  the  crown.  (Fig.  1).  Formulas  (d) 
and  (e)  are  used  in  finding  the  vertical  and  horizon- 
tal deflections  between  a and  c and  between  b and  c. 
As  the  supports  are  actually  fixed,  the  vertical  deflec- 
tion between  a and  i must  be  equal  to  that  between  b 
and  c;  thus  giving  the  first  condition.  Also,  the  horiz- 
ontal deflections  between  a and  c and  between  b and  c 
must  be  equal  in  magnitude  and  opposite  in  di- 
rection, because  what  is  lengthened  in  one-half  of 
the  rib  must  be  shortened  in  the  other.  Thus,  we 
obtain  the  second  condition.  For  the  left  half  of 
the  rib  with  the  origin  at  a the  moment  at  any  point 
■between  a and  c is 

M = rij -t\£a?  ~Hy  -P(~c-fd)  ••••(!) 

For  the  right  half  of  the  rib  with  the  origin  at  b 
the  moment  at  any  point  between  b and  c is 

M=Mzi-\t,3c-Hy  (2) 

Substituting  (1)  in  ( d ) and  ( e ) and  replacing  the 
integral  forms  with  the  notation  above  given,  we 
have  the  horizontal  and  vertical  deflections  be- 
tween a and  c respectively, 

EA.ir  - i- Vxr  ~/it  -P(r2-/ilq2)'  • (3) 
JEAy  =7^/?  i-lfs-Hr  -P(p^-kCp^  • • (i 

Substituting  (2)  in  the  same  formulas  and  making 
the  same  substitutions  for  the  integral  forms,  we 


P=P,  +P? 

q = + <ja 

r = r,+r2 


have 


Eaot  = Ptaq  fit  — (5) 

Paz/  - M^p  -t-  Ur  -...(6) 

where  the  modulus  of  elasticity  is  assumed  to  be 
constant.  Equating  (3)  to  (5)  and  (4)  to  (6)  as 
above  mentioned,  we  have 


S£Jkl 


^zpc/s 


r ^ 

~J/d  r 


£5  = S,  / 
t = t.  £s. 


prsq 


The  hinge  at  the  crown  breaks  the  continuity  of 
the  beam,  hence  a special  method  must  be  used 
in  applying  the  two  elastic  conditions.  M.  A.  Howe 
in  his  book  called  “A  Treatise  on  Arches”  succeeds 
in  reducing  the  required  elastic  conditions  into  one 
by  means  of  symmetrical  loading.  But  the  method 
is  rather  lengthy,  and  the  formulas  obtained  rather 
complicated.  By  means  of  symmetrical  deflection 
equations,  the  problem  is  easily  solved  by  the 
author,  with  the  final  formulas  expressed  in  a very 
simple  form.  Let  us  assume  that  the  arch-rib  >s 
divided  into  two  parts  at  the  crown  by  an  imaginary 


y- Ifs  -Hr  -P(k>  -k.  ZpP)  — -Hr  1 

-h  Ifr  -Ht  ~P(r3 -ftlqe)  = ~(j%  q +l£r-Ht) 

V (8) 
These  two  equations  together  with  the  three  statical 
equations  and  the  equation  furnished  by  the  hinge 
at  the  crown  as  shown  below  are  sufficient  to  solve 
all  the  unknowns. 

M,  ....  (9) 

/ T*L  / /l  • • • ( 10 ) 

j%  -a*  - m^sk)  -° 

V+\£-P=o 

H,  -H^  = o ...  (12) 

Thus,  by  solving  (7),  (8),  (9),  (10),  (11)  and  (12;, 
we  have 


4 


B'.-iZ-rMlL-ff-  (A) 

v _ „ Ip-kLpx-s-ss  (B) 

* Ip -as 

(C) 

Mj  = - Vi fj  + &(j-zk)  (D ) 

M2~  Ml  1-Jfl  PI  (l-fi ) (E ) 

It  must  be  remembered  that  these  formulas  hold 
good  only  when  the  load  is  on  the  left  of  the  crown. 
They  are  applicable  to  a one-hinged  arch  with  any 
form  of  arch-ribs. 


1 

F/gJ 
v.  c 
0 $ 


\Yz 


K- A- f — 

3 

fir 

Mis 

FO 

t i 

i 

F/g.2. 

These  formulas  again  hold  good  only  when  the 
load  is  on  the  left  of  the  crown  with  its  direction 
toward  the  left.  They  are  also  applicable  to  any 
form  of  arch-ribs. 

Reactions  Solved  by  Author’s  Cantilever  Method 

Another  interesting  method!  is  derived  by  the 
author  in  securing  the  two  elastic  equations  for  solv- 
ing the  reactions  of  a one-hinged  arch.  It  is  called 
the  cantilever  method,  because  the  main  feature  lies  in 
separating  the  arch  into  two  separate  cantilevers  free 
at  the  crown  and  fixed  at  the  supports.  The  vertical 
and  horizontal  deflections  at  the  free  end  of  each  can- 
tilever are  then  found  and  equated1  so  as  to  furnish 
the  two  necessary  equations.  The  method  is  very 
simple,  because  the  deflections  at  the  free  end's  of  can- 
tilevers can  be  easily  obtained.  Thus,  the  following 
deflection  table  includes  all  the  data  needed  for  the 
solution : 

Table  1.  Deflection  Table  for  Curved  Beams 


Load/ng 

D/agra/7? 

Ax.  at  c 

Ay  at  c- 

Vert  Lootof  of 
M from  <=> 

VLAR. 

- # H-  ^ 

hor  Load  at 
k!  from  o 

( -jr  (dkfi-r) 

Vert  Loacf  of 
free  enof 

_ |C 

-f 

~ ^ (ip-ri) 

Hor  Loo  of 
at  free  esief 

!/<  £ 

o\ i__i 

+ (hfi-  r) 

e-.J.s. 


If  instead  of  a vertical  load  P on  the  span,  we  place 
a horizontal  load  P at  the  same  position  acting 
towards  the  left  and  at  a distance  k'h  from  the  line 
ab,  (Fig.  2)  the  problem  can  be  solved  in  a similar 
way.  In  this  case  the  direction  of  II. > and  V2  are 
reversed  in  comparison  with  those  under  the  vertical 
loading.  The  following  formulas  are  derived  for 
the  horizontal  load  on  the  span  with  the  same 
process : 


«•  _ d shg-hk'g,-t~t 

fj, 

3 y ^ 

V -knpi 

as>  -ip 

m2=Mj  -jpfih 


....(F) 

....(G) 

....(H) 

.•••(I) 


The  expressions  in  this  table  can  be  easily  derived 

by  using  formulas  (d)  and  (e). 

/ 

In  order  to  find  the  elastic  conditions,  let  us  sepa- 
rate the  arch  at  the  crown.  Since  there  is  no  load  on 
the  right  half  of  the  rib  (See  Fig.  3)  and  since  the 
right  reaction  line  must  pass  through  the  crown,  the 
two  forces  H and  V2  must  act  at  the  free  end  of  the 
right  cantilever  with  their  directions  as  shown. 
Therefore  we  can  consider  the  left  cantilever  as  sub- 
jected to  two  external  forces  H and  V 2 at  the  free  end. 
On  the  left  half  of  the  rib  there  is  a load  P.  But 
since  action  must  equal  the  reaction,  the  forces  acting 
at  the  free  end  of  the  left  cantilever  must  be  H and 
V 2.  Let  ns  find  the  vertical  and  horizontal  deflections 
between  a and  c when  the  loads  H,  V2,  and  P are  act- 
ing on  the  left  cantilever.  Evidently,  the  vertical  and 
horizontal  deflections  at  c are  contributed  by  three 
factors,  those  due  to  P,  H and  V2  respectively.  From 
table  1 these  factors  can  be  summed  up  in  the  ex- 
pressions, 

J2jk.p  = -f-H(hp-r)  —p(klpi  -Sjj  -(13) 

=\^(Lq-r)  (14) 

Similarly,  the  vertical  and  the  horizontal  deflections 
at  c of  the  right  cantilever  are  contributed  by  two  fac- 
tors, those  due  to  H and  V 2.  From  table  1,  we  have 
the  expressions, 


5 


Table  2 — Reactions  and  End  Moments  of  a One-Hinged  Ar  h with  Various  Forms  of  Arch-Ribs. 


1 k/ 

P 0-2)1 

Si* 
/ -w 
/a 

< 

r l '' 

One-hinged  Arch 

m'V  K-K* 
n*  3z- 3 r* 

0 = 3tt-8 

oc=sin'(zk-/j 

g = r-h 

cosec-  ° T7- 

r-  +i£±lM 
th 

1 * I0  Sec  3 

E * Constant 


P 

r=  fiityds 

n-r^yf6 

S.jfxVs 
s, 

sA\^ 

K = zarfr-  §)(”  - oc)+l(f-izr-ahg) 
d « L2r*l-l3-z4rzg(j£ -oc) 

NOTE- 

/ Notations  as  shown  above.. 

Z.  Assumptions,  1 = 1°  Sec  e. 

E = constant,  shear  nea- 
lected 

3.  Formulae  under  Bending 
F)omenf  ho/d  good,  when 
the  toad  is  on  tn> 
of  the  span. 


CURVE: 


15 

I 

« 

§ 

CQ 


4j  K 

led 


e left  ha  If 


b <& 

5 § | 

u>  i:  ^ 
<o  >u 


H 


Vi 


14 


Mi 


Me 


H, 


Uz 


V, 


Vz 


Mi 


Me 


n, 


Mi 


Hi 


Ml 


GENERAL 


pkiy-'i  . 
2(h9-t) 


P bApi-s, 
Lp-ZS 


-V,±+tih-rQ(,-Zk) 


M,+V,L-Pl(l-k.) 


peho-k’hg.-t-t' 

Z(hq-t) 


pJdhOi-t, 


pJL-.h'frp, 
ZS  - Lp 


. p n - Ftp, 
Zs-lp 


vd  + h,h-Ph(l-k) 


M,+V,L  - Pk'h 


Set" l 


z^-tyin' 


^ Eet°lh 

if. 

ZLL- 


Z(hq-t)*-ln2 


- sL 

ZChq-t) 


shL 

2(h?-t) 


PARABOLIC 


#f  (***-**) 


pl  [«<ilk’-4k+i)in+ 
Tho\^  3 Q-Z k)(3 tt-zoc) 


P(/-4k3) 


4Pk  3 


-PLQek-i^fJfk) 


ELQsk’-sk*) 


sQ-2ok3+4ok?-r6ks) 


PP(5k-)ok‘)-4ks) 


lEb(zC-  3k*) 


- ePh(zk3-3k‘>) 


4Ph(k-k  - 7k) /3k? 4k) 


-4Ph(3k-VV-t4K) 


± iSFet'lo 

zh*+i5n 3 


zp  15  E<zt°Lh 
Zhz  + l5o* 


_ IS  slo 
Zhz 


L5sIo 


EzLLIPTICAL 


pQ-ok3) 


4-Pk 


-Vl+Hh+%Q-2k) 


-PlQ-k ) 


'o+4k(3-3ki-Zk') 

- 3m(z<c-3n ) 


P-  H , 


Ph  (4m(akfZkf3)-l 

til  ) 


-V, 


~{iL+  Hh  - PhQ-k) 

z 


M.-hVk-Pk'h 


!2  Set's. 

Oh  2 + 120* 


^ izEet'Lh 
oh*+  !2.r,z 


12  31° 

ohz 


IZSlo 

oh 


3EGMENTAL 

CIRCULAR 


. — 1 


ZMJ+ZK-Zoc)-Jf’. 


PQ-dk3) 


4Pk3 


M,  f til- PlQ-k) 


p-hz 


Zk  L [/2  r2- 1\3 -e*  i-  4 h 

U6r*fsir>24c'+2(c-*c)]\g+r6tni 

\6lrg(/-4K)sinoc' 


pjsiRecV^  sinZoc] 
[fekzsinoc'--§KJ 


U +d,h-Ph(l-k) 


M,fti,l~Pk'h 


± /z  Eet'll* 
K+izln* 


T izket°ll°h 
K + IZlr,* 


!2sl  I , 


y.  IZslhL 
H 


Table  3 — General  Formulas  of  Reactions  and  End  Moments  of  an  Arch  with  the  Regular  Form  of  Arch  Rib. 


P,ft*p 

r+ 

S - ft  4^ds 

Jo  l 

p*-ft  ¥6 

<1- find? 

szMj  ^ 

iMffyds 

t -/j 

q.- £nds 

tpjf'gtds 

n.Jf^s 

1 * I0  sec  e 

r2.fi*qds 

£ ■ ConsEarrb 

NOTE- 

/.  Formulae  for  one  and  three 
hinged  Arches  under  the 
term  Bending  Moment  hold 
good,  when  the  load  is  on  the 
left  of  C. 

2.  B/se  or  Fall  of  Temperature 
has  certain  effect  upon  M 
of  the  three  hinged.  The 
formula  is  not  derived,  as 
its  effect  upon  the  truss 
can  best  be  found  bu  the 
graphical  method,  d — ) 

3.  Notations  are  shown  above, 
r,  - radius  of  Gyration  s =■ 
unit  compressive  stress  in 
Pib-shortening  Formulae. 
(Average  stress) 


ARCH 

NO-HI  NGEzD 

ONE-HINGED 

TWO-NINGLD 

THEE  L -HINGE  D 

£ 

% 

? 

1 

Q 

§ 

-4 

O 

K 

0) 

s 

H 

Bl(brk.l<}2)-l<?(k-k') 

2 ir-2q‘L 

e>  kloi  — n 

Zfrq-t) 

a /7  + k.lQi. 
z-t 

P>k.l 

~zK 

V, 

+ F-Q-k) 

f=>  IfO  -kl)0,-S-S* 
lp-zs 

P(L-k) 

pQ-k) 

Vz 

P-V, 

p kip, -3, 
lp—2s 

Pk 

Pk 

Mi 

4^0  Qlq  -3d)-Pkl(/-kf 

-V/L-r  Hh  + P)  Q-Zk) 

o 

o 

Mz 

pr,  + Kl  -Pl(/-k) 

M,  +-V/L- PlQ-k) 

o 

o 

MOEUZON TA  L LOAD 

n 

dfa-k'hqj-qlzq,-  s 

S\  If  k'UQ-te) 1 > 

2 L it-z<r'/,  1 

p zhg-Fha,  -t-ti 

EQ  + tt  - kings) 

§(*-k) 

Hz 

p-n, 

p k’hat-t, 

Z&q-t) 

P-H, 

Pk.' 

z 

V, 

+ Pk'h) 

P n-k'hp, 

2s  —Lp 

Pk'h 

L 

Pk'h 

L 

1/a 

-V, 

-P  r,  - k'hp, 

2S-Lp 

- Pk'h 
l 

- Pk'h 
l 

M, 

%QL<t-3d)Af’])(r,-n)- 
Zl  (2<f-qi)+Pk  'h(l-4k+3k‘) 

-U  +H,h-Ph(i-k) 
z. 

o 

c 

Mz 

rv),  + V,L -Pk'h 

M,  fV,l -Pk'h 

o 

o 

Qi . jo 

Igg 

JU  b k 

K x uj 

H, 

. tzehl 

± Bei°l 

* Bet°l 

* 2-t - 4z±L  + in' 
i ~TZ 

Z&q-t)  * ifj 

M, 

-r-  2Be-t0n 

o 

o 

Sgs 

Q1  5 $ 

«5  ^ 

h, 

_ sl 

zb-  iff. 

— ^ s — 

KM-t) 

-Hr 

o 

M, 

+ jshl — , — 
Z(hg-t) 

o 

o 

EAy  = -%(lp  ~s)  +H(f?p-r)  ’ ‘ ( 15  j 

EJAa?  = -V^{Lg-r)  -Plf(hg-t)  • • • (16) 

The  value  of  E is  assumed  to  be  constant  in  all  of 
these  expressions.  Equating  the  vertical  deflections 
from  each  half  of  the  rib,  we  have 


= -IzCkp-'S)  -hH(hp-r)  ( ' 

from  which  the  formula  for  V2  can  be  obtained  di- 
rectly. Equating  the  horizontal  deflections  we  have 

+*f(/p-r)  -F>(klp,  -s,)' 

- v^kp-r)  -H(hq-t) (18) 


from  which  the  formula  for  H can  be  obtained  di- 
rectly. From  the  statical  conditions  and  the  condi- 
tion furnished  by  the  hinge  at  the  crown,  we  can 
obtain  all  necessary  formulas  for  the  reactions  and  the 
bending  moments  at  the  ends. 

In  the  case  of  the  horizontal  loading,  the  above 
method  applies  equally  well.  The  expressions  for  the 
vertical  and  horizontal  deflections  at  the  free  end  of 
a cantilever  under  a horizontal  load  P are  also  given 
in  table  1. 

Influence  of  Temperature 

Changes  in  temperature  cause  changes  in  the  values 
of  and  II i but  do  not  affect  the  vertical  reaction  l7^ 
It.  is  usually  specified  that  an  arch  shall  be  designed  to 
be  subject  to  a certain  variation  in  temperature  from 
a standard  value.  Let  2 be  the  coefficient  of  expan- 
sion and  r the  rise  of  temperature,  then  the  spau 
length  will  be  increased  by  %t°l  provided  one  end  is 
free  to  move.  As  both  ends  are  fixed  in  position  when 
the  supports  do  not  yield,  equal  and  opposite  reactions 
and  end  moments  are  produced.  The  values  of  II, 
and  1/j  must  be  such  as  to  prevent  the  horizontal  dis- 
placement 2t°£,  which  is  due  to  the  effect  of  bending 
and  thrust.  The  horizontal  deflection  due  to  bending 
and  thrust  are  given  by  the  following  expressions: 


in  which  A is  the  area  of  cross-section.  Equating  the 
deflections,  and 

substituting  H^h-y)  for  M and  solving  for  H1  and  Mu 
we  have 


■/.  Eet°l 


(K) 

(L) 


where  Io  is  the  moment  of  inertia  of  the  section  at  the 
crown  and  )\  is  the  radius  of  gyration  of  the  same  sec- 
tion. For  a rise  in  temperature  use  the  positive  sign 
for  II 1 and  a negative  sign  for  M\.  For  a fall  in  tem- 
perature, the  reverse  is  true. 

Rib-Shortening 

The  direct  effect  of  the  thrust  along  the  axis  is  to 
shorten  the  axis  of  the  arch-rib.  It  would  also  shorten 
the  span  provided  one  end  were  free  to  move,  but  as 
this  is  not  the  case  it  will  develop  equal  and  opposite 
negative  reactions  H1  and  positive  moments  M\.  The 
effect  of  displacement  due  to  II 1 and  M\  must  equal 
that  due  to  rib-shortening.  Let  us  use  S as  the  aver- 
age compressive  stress  on  the  rib.  The  shortening  of 
the  span  is  given  by 


A. 


Si  (21) 

E 


which  must  be  equal  to 


(22) 


Equating  (18)  to  (19),  substituting  H^h-y)  for  M in 
the  equation,  and  solving  for  H u we  have 


r=_  SI 

a(hg  - 1) 


Shi 

a(hg-t) 


(M) 

(N) 


Application  to  Parabolic,  Segmental  Circular,  and 
Elliptical  One-Hinged  Arches 

The  general  formulas  derived  above  have  been  ap- 
plied to  three  forms  of  arch-ribs  with  parabolic,  seg- 
mental circular  and  elliptical  curves  respectively.  The 
equations  of  these  curves  all  have  their  origin  at  the 
left  support  a,  with  the  X-axis  passing  through  the 
supports  a and  b.  The  equations  of  these  curves  are 
as  follows: 


(y  -f-r-hf  = ) • (24) 


An  important  assumption  is  made  in  applying  the 
general  formulas,  that  is  I=lo  sec  e where  7 is  the 
moment  of  inertia  of  the  cross-section  at  any  point  and 


7 


o the  angle  of  inclination  of  the  axis  of  rib  at  the  point 
where  / is  taken.  This  assumption  simplifies  the  work 
materially  and  should  be  dose  enough  for  deriving 
the  formulas  to  be  used  in  the  preliminary  designs. 
Also,  ds=sec  a dx,  while  the  modulus  of  elasticity  is 
assumed  to  be  constant  in  the  general  formulas. 
Table  2 gives  a comparison  of  the  formulas  for  the 
reactions  and  end  moments  of  the  one-hinged  arches 
with  parabolic,  elliptical  and  segmental  circular  arch 
ribs.  These  formulas  have  been  derived  after  a tre- 
mendous amount  of  work.  Formulas  for  tempera- 
ture effects  and  rib-shortening  are  also  given  in  the 
table. 

The  Reaction  Locus 

Fig.  1 shows  a one-hinged  arch  under  a vertical 
load  P.  As  the  ends  are  fixed,  it  is  unknown  where 
the  reaction  lines  must  pass  the  supports.  Let  us 
assume  that  they  pass  the  left  and  the  right  supports 
at  points  distant  y1  and  y2  below  and  above  the  points 
of  support  respectively.  The  right  reaction  line  must 
pass  through  the  center  hinge.  By  similar  triangles, 
we  have  the  relation 


in  which  q is  the  ordi- 

kl  H 


nate  to  the  curve  at  any  point 
But 


IJt  = > therefore 


fj filVi  -/-I'll 

^ III 


(0) 


which  is  a general  formula  of  the  reaction  locus  of  the 
one-hinged  and  the  no-hinged  arches  applicable  to 
any  form  of  arch-rib.  It  is  not  applicable  when  the 
load  is  on  the  right  half  of  the  arch  rib,.  The  reaction 
locus  must  be  symmetrical  about  the  center,  so  it  is 
not  necessary  to  d'erive  the  formulas  for  the  load  on 
the  right  of  the  span.  For  parabolic,  elliptical  and 
segmental  circular  arches  we  have  by  substitution  the 
following  formulas : 


h(/4  -/3k) 


9 


h(/+/3k)  .(P) 

3(/ik)~ 


for  the  load  on  the  left  and  right  of  the  crown  hinge 
respectively. 

Equation  Q* 

Equation  Rf 


Formulas  for  Reactions  of  Three — ,Two — And 
No-Hinged  Arches 

General  formulas  for  reactions  of  three-,  two-, 
and  no-hinged  arches  have  been  derived  in  order  to 
obtain  a comparison  between  them.  They  have  also 
been  applied  to  these  types  of  arches  with  parabolic, 
elliptical,  and  segmental  circular  ribs.  Table  3 
shows  a comparison  of  the  general  formulas  of  the 
four  types  of  arches,  while  those  for  a comparison 
of  the  reactions  of  parabolic,  segmental  circular  and 
elliptical  forms  of  arches  are  not  here  reproduced. 

The  Envelop  of  Reaction  Lines 

The  position  of  live  load  for  the  maximum  posi- 
tive and  negative  moments  and  the  maximum  posi- 
tive and  negative  shears  of  the  one-hinged  arch  can 
be  found  graphically  by  means  of  the  reaction  locus 
and  the  envelop  of  the  reaction  lines.  Since  the 
ends  of  the  one-hinged  arch  are  fixed,  we  cannot 
tell  where  the  reaction  lines  pass  through  the  sup- 
port for  a load  at  a certain  point  of  the  span.  By 
means  of  the  reaction  envelop  and  the  reaction  locus, 
this  can  be  easily  done  by  drawing  the  tangent  line 
to  the  reaction  envelop  from  the  point  where  the 
load  line  intersects  the  reaction  locus.  The  equa- 
tion of  the  reaction  envelop  of  the  one-hinged  arch 
was  found  by  the  author  to  be  an  equation  of  the 
third  degree.  To  plot  the  curve  from  the  equation 
involves  a tremendous  amount  of  labor.  It  is  easier 
to  draw  the  reaction  envelop  from  the  computed 
values  of  ordinates  than  to  plot  it  from  the  equation. 
The  envelop  passes  through  the  hinge  and  intersects 
the  Y-axis  at  a certain  distance  below  the  origin. 
To  determine  the  points  of  division  of  the  live  load 
for  the  maximum  positive  and  negative  moments, 
tangent  lines  are  draAvn  to  the  envelop  from  the 
point  on  the  arch-rib  we  wish  to  investigate.  The 
intersection  of  the  tangent  lines  with  the  reaction 
locus  gives  the  required  points  of  division.  To  de- 
termine the  points  of  division  for  the  shear,  a 
tangent  line  to  the  envelop  is  drawn  perpendicular 
to  the  normal  line  of  the  section.  The  intersection 
of  this  tangent  line  with  the  reaction  locus  forms  one 
division  point,  while  the  section  itself  is  another 
division  point. 

Study  On  Designs 

The  design  of  a one-hinged  arch  can  only  be  made 
with  a series  of  approximations.  Stresses  due  to 
temperature  and  rib-shortening  play  an  important 


q=h^/+8(37r-3) 


(/—2ft) ft3 "1 

4(4 ke -3- ft  -P3)-)k-ff-h3(/-2K)(37r  -3a ) J 


in  which 


Ct^cos 


t 


1-h 


4 Ik  *(/-2./t) 


h[sr3  &i7z3c*'j-3rsl(/-£fi)(stvi  £o<'-/-2ci-2c(')-  8 g3^3l^(j -3 ft  +4ft2^ 

in  which  Cf  = r~h  tx'—  siii~I(c±k.-/') 


(Q) 


(R) 


8 


part.  These  can  not  be  exactly  determined  without 
knowing  the  area  of  the  cross-section  of  the  arch- 
rib.  Yet  the  latter  again  depends  upon  the  total 
stresses  to  be  carried.  The  method  of  trial  is  the 
only  way  to  secure  the  right  section  for  the  one- 
hinged  arch-rib.  Three  comparative  designs  were 
made  by  the  author  in  order  to  study  their  compara- 
tive merits:  (1)  The  moment  of  inertia  at  any 
section  of  the  arch-rib  is  assumed  to  vary  as  the 
secant  of  the  angle  of  inclination  of  the  section, 
with  the  effect  of  rib-shortening  neglected;  (2) 
Same  assumption  as  in  the  first  case,  with  the  effect 
of  rib-shortening  included;  and  (3)  the  true  moment 
of  inertia  of  the  section  is  used. 

A one-hinged  arch  of  twenty  panels,  with  a span 
of  258  feet  and  a rise  of  26'  was  adopted  for  investi- 
gation. The  dead  load  is  assumed  59  kips  and  the 
live  load  18.5  kips  per  panel  load.  The  depth  of  the 
rib  at  the  center  is  assumed  5 feet.  An  allowance 
of  75°  F is  made  for  the  rise  or  fall  of  temperature. 
The  Modulus  of  elasticity  is  taken  as  26,000  kips  and 
the  unit-stress  for  the  gross  section  of  the  flanges, 
15  kips.  The  arch  chosen  for  investigation  has  been 
greatly  studied  by  former  graduate  students  of  Pro- 
fessor H.  S.  Jacoby  when  containing  three,  two  or 
no  hinges.  The  reason  for  using  this  arch  by  the 
author  is,  of  course,  to  compare  the  design  of  the 
one-hinged  arch  with  that  of  the  other  types. 

Study  On  Deflections 

The  deflections  of  the  one-hinged  arch  wTere  stud- 
ied in  three  ways:  (1)  Vertical  and  horizontal  de- 
flections under  the  vertical  loading ; (2)  vertical 
and  horizontal  deflection  under  the  horizontal  load- 
ing; and  (3)  maximum  and  minimum  deflections. 
The  same  arch  used  in  the  design  was  used  for  the 
study  of  deflections.  The  deflections  of  this  arch 
having  three,  two  and  no  hinges  respectively  were 
investigated  by  Dr.  P.  H.  Chen,  a graduate  of  Cor- 
nell University.  It  gives  a good  opportunity  for 
critical  comparisons.  (See  The  Cornell  Civil  Engi- 
neer, Vol.  26,  Page  229). 

RESULTS  OF  INVESTIGATION 
Discussion  On  Formulas 

So  far  as  the  formulas  for  the  reactions  of  the  one- 
hinged  arch  are.  concerned,  those  for  the  parabolic 
ribs  are  the  simplest  in  form,  while  those  for  the 
segmental  circular  form  require  a vast  amount  of 
labor  for  their  derivation  and  application.  Formu- 
las for  the  elliptical  form  are  intermediate  in  these 
respects. 

Neglecting  the  effect  of  the  axial  thrust  and  using 
a ratio  of  the  rise  to  the  span  for  a segmental  cir- 
cular arch  equal  to  1-8,  the  relative  effect  of  tem- 
perature upon  the  three  forms  of  the  one-hinged 
arch  is  as  follows : 

Seg. 

Parabolic  Elliptical  Circular 
Coefficient  of  Eet°Io  /;2  7.5  8.4  6.0 

Ratio  125  140  100 


The  temperature  effect  in  the  parabolic  form  is 
125%  as  great,  and  that  in  the  elliptical  form  140% 
as  great  as  that  for  the  segmental  circular  form. 
The  effect  upon  the  elliptical  form  is  112%  as  great 
as  that  upon  the  parabolic  form.  Thus,  the  tem- 
perature effect  upon  the  elliptical  form  is  the  larg- 
est, while  that  upon  the  circular  form  is  the  small- 
est. It  is  to  be  noted  that  in  the  segmental  form 
the  ratio  of  the  rise  to  the  span  is  assumed  as  one- 
eightli.  This  is  only  a particular  case.  If  we  as- 
sumed another  ratio,  the  results  would  be  different. 

The  relative  effects  of  rib-shortening  upon  the 
horizontal  reaction  and  the  end  moments  of  the 
three  forms  of  one-hinged  arches  have  the  same  re- 
lations as  the  temperature  effects. 

There  is  an  interesting  fact  about  the  number  of 
equations  required  to  analyze  no-,  one-,  two-, 
and  three-hinged  arches.  Three-hinged  arches  re- 
quire three  conditions;  two-hinged  arches,  four;  one- 
hinged  arches,  five ; and  no-hinged  arches,  six.  Thus, 
we  see  that  the  more  hinges  an  arch  has,  the  less 
are  the  number  of  the  conditions  required  to  solve 
its  unknowns.  In  addition  to  the  three  statical  con- 
ditions, the  three-hinged  arch  requires  no-elastic 
condition  for  its  solution ; the  two-hinged  arch  re- 
quires one ; the  one-hinged  arch,  two ; while  the  ne- 
hinged  arch  requires  three.  Therefore  the  three- 
hinged  arch  is  said  to  be  statically  determinate. 

The  formulas  for  reactions  of  three-hinged  arches 
are  exact,  while  those  for  no-,  one-,  and  two- 
hinged  arches  are  subject  to  many  imperfections 
and  assumptions.  First  of  all,  the  elastic  limit  is 
not  assumed  to  be  passed.  If  very  large  loads 
should  ever  be  applied  which  cause  the  stresses  to 
exceed  the  elastic  limit  of  the  material  in  any  mem- 
ber, the  theory  of  elasticity  fails  and  it  is  impossible 
to  predict  the  degree  of  safety  of  the  structure.  The 
moment  of  inertia  of  any  section  of  the  rib  is  as- 
sumed to  vary  as  the  secant  of  the  angle  of  the 
inclination  of  the  arch-rib.  The  modulus  of  elastic- 
ity is  assumed  to  be  constant  throughout  the  span. 
Shear  is  neglected  in  the  derivation  of  the  formulas. 

No-  and  one-hinged  arches  are  similar  in  one  re- 
spect; that  is,  they  are  both  subject  to  vertical  and 
horizontal  reactions  and  have  end  moments  under 
either  vertical  or  horizontal  loads.  Two-  and  three- 
hinged  arches  are  all  subject  to  vertical  and  horizon- 
tal reactions,  the  vertical  reactions  being  the  same 
for  those  two  types  of  arches  under  either  vertical  or 
horizontal  loads.  Temperature  and  rib-shortening 
cause  horizontal  reactions  and  the  end  moments  in 
both  the  one  and  no-hinged  arches,  while  in  two- 
hinged  arches  they  cause  horizontal  reactions  but 
no  end  moments.  It  is  generally  supposed  that  a 
three-hinged  arch  is  not  subject  to  stresses  due  to  a 
change  in  temperature.  Strictly,  however,  such 
stresses  will  occur,  for  a fall  in  temperature  causes 
a deflection  of  the  crown  hinge,  and  as  the  span 
does  not  change,  the  horizontal  thrust  will  be  in- 


9 


creased.  The  stresses  produced,  however,  are  very 
small. 


Neglecting  the  effect  of  axial  thrust  upon  the  rise 
and  fall  of  temperature,  the  relative  importance  of 
temperature  effects  upon  the  horizontal  reactions 
and  end  moments  of  no,  one-,  twro-,  and  three- 
hinged  arches  are  as  follows : 


Arch 

No- 

One- 

Two- 

Three- 

Hinged 

Hinged 

Hinged 

Hinged 

ELLIPTICAL 

160/8 

67.2/8 

12/8 

0 

Mx 

125.6/8 

67.2/8 

0 

0 

PARABOLIC 

Hx 

90/8 

60/8 

15/8 

Mx 

60/8 

60/8 

0 

0 

Thus,  the  temperature  effect  upon  the  no-hinged 
arch  is  found  to  be  the  greatest  of  all.  Referring 
to  H in  the  parabolic  form,  it  is  one  and  one-half 
times  as  large  as  for  the  one-hinged  arch,  and  six 
times  as  large  as  for  the  two-hinged  arch.  In  the 
elliptical  type  it  is  2.38  times  as  large  as  that  for  the 
one-hinged  arch,  and  13.33  times  as  large  as  for  the 
two-hinged  arch.  The  temperature  effect  upon  H of 
the  one-hinged  arch  of  the  parabolic  form  is  four 
times  as  large  as  for  the  two-hinged  arch,  while  for 
the  elliptical  form  it  is  5.6  times  as  large.  The  tem- 
perature effect  upon  M of  the  no-  and  one-hinged 
arches  is  the  same  in  the  parabolic  form.  In  the 


elliptical  form,  the  former  is  18.7  times  as  great  as 
the  latter.  In  the  segmental  circular  form,  the  tem- 
perature effect  upon  H in  the  one-hinged  arch  is 
three  times  as  great  as  that  for  the  two-hinged 
arch. 


With  respect  to  the  effect  of  rib-shortening  the  re- 
lative ratios  above  stated  apply  to  different  types 
of  arches  equally  well. 

So  far  as  the  formulas  for  the  reactions  are  con- 
cerned, there  is  a decided  advantage  gained  by  those 
for  the  parabolic  form  over  all  other  types  of  arches 
because  of  their  simplicity. 


Discussion  on  Reaction  Influence  Lines 

In  order  to  study  the  variation  of  the  reactions 
and  the  end  moments  as  a single  load  moves  over 
the  span,  reaction  influence  lines  are  drawn  and 
compared.  (See  diagrams  2,  3,  4,  8,  9,  and  10). 
Only  a few  of  the  numerous  diagrams  are  repro- 
duced in  the  abstract  of  the  thesis. 

Under  vertical  loading,  the  elliptical  form  of  the 
one-hinged  arch  has  greater  horizontal  reactions  and 
end  moments  than  the  parabolic  and  segemental 
circular  forms.  The  last  two  forms  cause  less  dif- 
ference in  the  Ht  and  Mx.  The  vertical  reactions 
are  all  the  same  in  the  one-hinged  arch  for  the  three 
forms  of  ribs.  There  is  not  much  difference  shown 
under  the  horizontal  loading.  The  circular  form 
seems  to  have  the  greatest  advantage  for  the  one- 


10 


hinged  arch  judging  from  a comparison  of  the 
curves.  However,  the  curves  for  the  circular  form 
are  only  drawn  for  a single  case  with  a ratio  of  rise 
to  span  equal  to  1-8.  If  other  ratios  were  used,  the 
results  may  be  different.  On  the  other  hand,  the 
parabolic  form  has  the  advantage  of  low  reactions 
besides  that  of  simplicity  in  formulas.  Therefore, 
the  author  comes  to  the  conclusion  that  the  para- 
bolic curve  is  the  best  form  for  the  neutral  axis  of 
the  arch-rib  to  be  used  for  a one-hinged  arch. 


Under  vertical  loading  the  reaction  influence  line 
of  H for  a three-hinged  arch  consists  of  straight 
lines ; for  a two-hinged  arch,  a curve  resembling  a 
parabola ; for  a no-  and  a one-hinged  arch,  curves 
symmetrical  at  the  center.  Curves  for  one-  and 
three-hinged  arches  break  at  the  crown  while  those 
for  no — and  two-hinged  arches  do  not.  This  is  be- 
cause the  hinge  at  the  crown  of  the  one-  and  three- 
hinged  arches  breaks  the  continuity  of  the  rib. 
Curves  of  no-,  two-,  and  three-hinged  arches  are 
very  close  to  each  other,  while  that  for  the  one- 
hinged  arch  has  a great  variation,  being  lower  at 
the  supports  and  higher  near  the  crown.  This  brings 
out  one  disadvantageous  fact  for  the  one-hinged 
arch.  When  the  load  is  at  the  crown,  the  value  of 
H is  about  two  times  as  much  as  that  for  the  other 
types.  This  means  that  the  supports  have  to  resist 
a greater  horizontal  thrust,  and  extra  care  has  to  be 
taken  in  making  the  end  supports  secure.  The  end 


moment  of  the  one-hinged  arch  has  a greater  varia- 
tion than  that  of  the  no-hinged  arch. 


of  fk&cfar?  Zoc/ 
S//s/?Acr&/  & fnx/Arr 


The  one-hinged  arch  has  an  advantage  over  the 
no-hinged  arch  with  regard  to  the  effect  of  tempera- 
ture and  thrust,  but  it  is  not  so  favourable  with  re- 
gard to  the  reactions  and  the  end  moments  under 
vertical  loads.  For  reactions  due  to  temperature, 
thrust  and  vertical  loading,  the  one-hinged  arch  is 
not  as  advantageous  as  the  two — and  three-hinged 
arches.  So  far  as  the  simplicity  of  the  formulas  for 


fc‘srr/a?/-*sas?  of~  Ahfefao  Zoc/ 

Os?&,  /mo  <5* 

Ztsae  /? styes'  asx/er  & Mr*/  /&&/ 

Ctsrya  SA&e/  2 / 


the  reactions  is  concerned,  the  same  statement  holds 
true.  However,  attention  will  be  called  subsequent- 
ly to  some  decided  advantages  of  the  one-hinged 
arch. 

Discussion  On  Reaction  Loci 

In  order  to  compare  the  reaction  loci  of  different 
types  of  arches,  the  equations  of  the  reaction  loci 
for  no-,  one-,  two-,  and  three-hinged  arches  are 
plotted  for  different  forms  of  arch  ribs.  They  are 
shown  in  diagrams  20  and  21. 

All  reaction  loci  for  the  one-hinged  arch  with  dif- 
ferent forms  of  arch-ribs  pass  through  the  crown 
hinge.  When  the  load  is  at  the  crown,  the  reaction 
lines  must  pass  through  the  crown  hinge,  for  other- 
wise there  would  be  rotation  at  the  crown.  All  re- 
action loci  break  their  continuity  at  the  crown  but 
the  two  halves  are  symmetrical.  The  parabolic  one- 
hinged  arch  has  the  highest  ordinate  when  the  load 


11 


is  near  the  support,  while  the  segmental  circular 
form  has  the  least  ordinate. 

For  a parabolic  arch-rib,  the  reaction  locus  of  the 
three-hinged  arch  is  a straight  line  for  each  half  of 
the  span;  that  for  the  two-hinged  arch,  a curve  like 
a parabola;  that  for  a no-hinged  arch,  a horizontal 
straight  line  at  an  ordinate  of  1,2  h;  and  that  for 
the  one-hinged  arch,  a curve  breaking  at  the  center. 
Curves  for  no-  and  two-hinged  arches  are  contin- 
uous, while  those  for  the  other  two  break  at  the 
center.  Curves  for  no-  and  two-hinged  arches  do  not 
pass  through  the  crown,  while  those  for  the  one-  and 
three-hinged  arches  must  do  so.  At  the  support,  the 
three-hinged  arch  has  a maximum- ordinate  of  2.00h; 
the  two-hinged  arch,  1.60  h;  the  one-hinged  arch, 
1.40  h;  and  the  no-hinged  arch,.  1.20  h.  At  the 
crown  the  curve  for  the  two-hinged  arch  has  the 
highest  ordinate. 

Similar  statements  can  be  drawn  for  the  reaction 
loci  for  the  elliptical  forms  of  arches.  The  ordi- 
nates are  highest  in  certain  cases  and  lower  in 
others,  while  the  general  forms  are  about  the  same. 

Discussion  On  Reaction  Envelop 

By  means  of  the  reaction  locus  and  the  reaction 
lines,  the  position  of  live  loading  for  the  maximum 
positive  and  negative  moments  and  maximum  posi- 
tive and  negative  shears  for  the  one-,  two-,  three, 
and  no-hinged  arches  can  be  found  graphically.  In 
the  three-  and  two-hinged  arches,  the  reaction  lines 
can  easily  be  drawn  as  soon  as  the  position  of  the 
live  load  is  given;  because  the  hinges  at  each  sup- 
port fix  the  direction  of  the  reaction  lines.  In  the 
one-  and  no-hinged  arches,  the  reaction  lines  cannot 
be  so  easily  drawn,  if  the  position  of  the  load  is  only 
given ; for  the  fixing  of  the  supports  makes  the  di- 
rection of  the  reaction  lines  uncertain.  In  order  to 
avoid  the  difficulty,  reaction  envelops  are  required 
for  both  the  one-  and  no-hinged  arches. 

The  reaction  envelop  of  the  no-hinged  arch  was 
found  by  A.  V.  Saph,  a graduate  of  Cornell  Uni- 
versity, to  be  two  hyperbolas  symmetrical  and  tang- 
ent to  each  other  at  the  center  at  a point  2-3  h above 
the  level  of  the  supports.  The  reaction  envelop  of 
the  one-hinged  arch  was  found  by  the  author  to 
consist  of  two  curves  very  similar  to  those  for  the 
no-hinged  arch  in  form  yet  quite  different  in  their 
equations.  The  curves  are  symmetrical  and  meet 
each  other  at  the  crown  hinge.  The  reaction  en- 
velop is  a plane  curve  of  the  third  degree ; because 
the  formulas  of  the  reactions  of  the  one-hinged  arch 
involves  the  k’s  in  the  fourth  power.  In  the  case 
of  the  no-hinged  arch,  the  k's  in  the  formulas  of  the 
reactions  are  of  third  power;  hence  a second-degree 
curve  is  obtained  for  the  envelop. 

The  method  of  finding  the  positions  of  live  loading 
is  the  same  for  the  four  types  of  arches,  except 
that  the  reaction  envelops  of  the  no-  and  one-hinged 
arches  have  the  function  of  the  end  hinges  in  the 


case  of  the  three-  and  two-hinged  arches.  A com- 
parison of  the  reaction  envelops  of  the  no-  and  one- 
hinged  arches  is  given  in  diagram  28. 

Discussion  On  Design  With  the  Moment  of  Inertia 
Varying  As  the  Angle  of  Inclination  of  the  Arch-Rib 

There  is  no  marked  difference  in  the  procedure  of 
designing  the  arch-rib  for  the  two-,  one-,  and  no- 
hinged  steel  arches  with  the  assumption  that  I 
varies  as  sec  b . The  procedure  may  be  generalized 
in  the  following  heads:  (1)  calculation  of  reactions 
and  end  moments  from  the  formulas  for  a unit  load 
at  each  panel  point;  (2)  to  find  the  position  of  live 
loading  for  maximum  positive  and  negative  mo- 
ments, by  either  algebraic  or  graphical  methods; 
(3)  calculation  of  dead  load  and  live  load  moments 
for  each  section;  (4)  calculation  of  dead  load  and 
live  load  thrust;  (5)  calculation  of  moments  and 
thrust  due  to  temperature  effect  by  assuming  a cer- 
tain moment  of  inertia  at  the  crown;  (6)  calculation 
of  moments  and  thrust  due  to  effect  of  rib-shorten- 
ing by  means  of  the  assumed  moment  of  inertia  ; 
(7)  design  of  the  flange  area  of  the  crown  section 
using  the  maximum  moments  and  thrusts  so  obtain- 
ed; (8)  test  the  moment  of  inertia  of  the  crown 
section,  and  see  if  the  assumed  value  of  the  moment 
of  inertia  at  the  crown  section  agrees  with  the  value 
obtained;  (9)  after  the  right  value  of  I0  is  obtained 
the  flange  areas  at  other  sections  can  be  calculated 
by  using  the  relation  that  / = It  sec  Q ; (10)  the 
position  of  live  load  for  maximum  positive  and  nega- 
tive shear  is  obtained  by  either  graphical  or  alge- 
braic method,  the  maximum  shear  at  each  section  is 
secured  by  combining  the  dead  load,  live  load,  tem- 
perature and  rib-shortening  shears,  and  the  webs  are 
designed  accordingly.  The  design  of  the  three- 
hinged  arch  may  be  carried  out  in  the  similar  or- 
der with  the  exception  that  no  trial  is  required  in 
securing  the  sections. 

There  is  an  interesting  fact  found  in  combining 
the  moments  and  thrusts  caused  by  the  dead  load, 
live  load,  temperature  and  rib-shortening  to  produce 
the  maximum  stress  on  the  sections  of  the  one- 
hinged  arch.  The  live  load  may  be  considered  in 
three  ways;  (1)  loading  for  maximum  positive  mo- 
ments; (2)  loading  for  the  maximum  negative  mo- 
ments; (3)  loading  for  maximum  thrust.  In  the 
design  with  the  effect  of  rib-shortening  neglected, 
case  (1)  controls  for  the  section  0-4  and  case  (3) 
controls  for  the  section  5-10.  In  the  design  with 
the  effect  of  the  rib-shortening  included,  case  (2) 
controls  for  the  section  0-4,  while  case  (3)  still  con- 
trols for  the  sections  5-10.  The  reason  is  quite 
clear;  for  in  the  sections  near  the  center,  the  mo- 
ments are  comparatively  small,  while  the  thrusts 
are  large.  This  means  that  a full  live  load  is  re- 
quired to  design  the  sections  near  the  center.  As 
there  exist  large  bending  moments  in  the  sections 
near  the  support,  naturally  the  moments  control 


12 


the  design  for  those  sections.  The  effect  of  rib- 
shortening is  to  produce  the  negative  moments  and 
thrusts.  This  is  why  the  negative  moments  control 
the  end  sections  when  the  effect  of  the  rib-shorten- 
ing is  included.  The  design  of  the  one-hinged  arch 
is  different  from  that  of  the  two-,  and  no-hinged 
arch  in  two  respects.  The  design  of  the  center  sec- 
tions of  the  one-hinged  arch  is  controlled  by  the 
thrust  while  that  of  the  two-  and  no-hinged  arches 
is  controlled  by  the  moments.  The  reason  is  be- 
cause the  no-hinged  and  two-hinged  arches  have 
large  moments  at  the  middle  sections,  while  the 
one-hinged  arch  has  large  thrusts.  The  signs  of 
the  moments  and  thrusts  produced  by  the  effect  of 
temperature  and  rib-shortening  are  the  same  for  all 
sections  of  the  one-hinged  arch,  while  those  for  the 
no-hinged  arch  are  the  same  for  the  sections  below 
2-3  h of  the  rib,  and  opposite  each  other  for  the 
sections  above  2-3  h of  the  rib.  The  signs  of  the 
moments  and  thrusts  produced  by  the  effect  of  tem- 
perature and  rib-shortening  are  all  opposite  for  the 
two-hinged  arches. 

In  designing  the  sections  at  the  crown,  only  thrust 
is  used  for  the  one-hinged  arch.  This  makes  the 
design  of  the  one-liinged  arch  easier,  because  the 
right  value  of  the  moment  of  inertia  can  be  secured 
immediately. 

Positive  shear  governs  the  design  of  the  web  for 
all  types  of  arches,  whether  the  effect  of  rib-short- 
ening is  included  or  not.  The  latter  which  tends  to 
produce  negative  thrust,  tends  to  increase  the  posi- 
tive shear  in  all  sections  for  the  one-hinged  arch. 

Under  maximum  loading  the  comparative  effects 
of  dead  load,  live  load,  temperature  and  rib-short- 
ening on  the  flanges  are  shown  in  the  following 
table  and  also  on  curve  sheet  22 : 


Dead 

Live 

Tempera- 

Rib- 

Section 

load 

load 

ture 

shortening 

Per  Cent.  Per  Cent. 

Per  Cent. 

Per  Cent. 

0 

31.5 

29.7 

24.2 

14.6 

1 

38.8 

24.2 

23.0 

14.0 

2 

47.4 

17.7 

21.7 

13.2 

3 

58.2 

10.6 

19.3 

11.9 

4 

72.7 

1.3 

16.1 

9.9 

5 

75.8 

23.7 

20.6 

—20.1  ' 

6 

75.9 

23.8 

15.5 

—15.1 

7 

76.0 

23.8 

11.4 

—11.1 

8 

76.0 

23.9 

8.5 

— 8.4 

9 

76.0 

23.8 

6.6 

— 6.5 

10 

75.8 

23.8 

6.3 

— 5.9 

It  is 

seen  from  the 

curve  that  the  dead  load  has 

the  greatest  effect  of 

all,  and 

its  effect 

on  the  sec- 

tions  near  the  center  is  greater  than  those  near  the 
ends.  In  the  first  place,  this  is  because  the  dead 
panel  load  is  comparatively  great,  and  naturally  it 
takes  greater  stress.  In  the  second  place,  the  effect 
of  temperature  and  rib-shortening  near  the  center 
sections  is  greater  than  their  effect  on  the  sections 


near  the  ends.  This  makes  the  percentage  of  the 
stress  carried  by  the  dead  load  gradually  decrease 
in  the  sections  near  the  ends.  The  live  load  is  the 
next  important  factor  in  causing  flange  stresses. 
The  effect  of  temperature  and  rib-shortening  can 
never  be  omitted.  They  have  about  the  same  effect. 
Temperature  has  the  greater  effect  in  the  sections 
near  the  ends  than  in  those  near  the  center.  The 
same  is  true  for  the  rib-shortening.  The  negative 
signs  of  the  rib-shortening  in  the  sections  near  the. 
ends  are  explained  by  the  fact  that  both  negative 
thrusts  and  moments  are  used. 

Under  maximum  loading  the  comparative  effects 
of  moments  and  thrusts  on  the  flanges  is  shown  in 
the  following  table  and  also  shown  on  curve  sheet 


> : — 

Section 

Moment 

Thrust 

Per  Cent. 

Per  Cent. 

0 

73.1 

26.9 

1 

67.4 

32.6 

2 

60.3 

39.7 

3 

51.8 

<M# 

GO 

4 

43.1 

56.9 

5 

2.5 

97.5 

6 

1.5 

98.5 

7 

.7 

99.3 

8 

.4 

99.6 

9 

.1 

99.9 

10 

.0 

100.0 

One  apparent  conclusion 

can  be  drawn 

from  the 

curve ; that  is,  the  moment  has  far  greater  effect  on 
the  sections  near  the  ends  than  the  thrust,  while 
the  thrust  has  far  larger  effect  on  the  sections  near 
the  middle  than  the  moment.  The  effect  of  moment 
on  the  sections  5-10  is  practically  nothing.  As  a 
whole,  the  thrust  is  more  important  than  the  mo- 
ment. This  peculiar  fact  is  a special  feature  of  the 
one-hinged  arch. 

The  comparative  areas  of  the  designs  with  the  as- 
sumption / 5 sect*  with  rib-shortening  included  and 
neglected  are  shown  in  the  following  table  and  on 
curve  sheet  24 : — 


Rib-Shortening 

Rib-Shortening 

Section. 

Neglected. 

Inc. 

Area  in  sq.  in. 

Area  in  sq.  in. 

0 

166.69 

153.70 

1 

142.77 

127.00 

2 

121.39 

103.60 

3 

105.00 

84.00 

4 

88.84 

67.75 

5 

82.15 

67.70 

6 

77.39 

66.80 

7 

73.72 

66.00 

8 

71.10 

65.45 

9 

69.38 

64.95 

10 

68.71 

64.71 

It  is  seen 

from  the  curve 

that  the  one-hinged 

arch  requires  far  larger  areas  in  the  sections  near 
the  ends  than  the  sections  near  the  crown.  This  is 


13 


due  to  the  presence  of  the  large  end  moments.  It 
is  also  seen  that  the  effect  of  rib-shortening  is  to 
decrease  the  areas  of  sections.  We  thus  see  that  for 
this  particular  rise  of  span,  the  effect  of  rib-shorten- 
ing can  be  safely  but  not  economically  neglected  in 
the  design.  The  author  believes  that  this  is  equally 
true  for  other  ratios  of  rise  to  span,  because  in  any 


So  far  as  the  areas  of  sections  in  the  flanges 
are  concerned,  the  three-hinged  arch  is  the  most 
favorable  of  all,  while  the  no-hinged  and  one-hinged 
arch  require  the  largest  areas.  The  areas  of  the 
two-hinged  arch  are  intermediate  between  them. 
The  three-hinged  arch  requires  the  largest  area  at 
the  quarter  points;  for  the  moments  at  those  sec- 


V 60 
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Comparative  Effects  of  D.L.  L.L. 
77  <3  R.S.  on  the  Flanges 


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Comparative  Web  Frees  of  0-R, 
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Comparative  Effects  of  Moment 
and  Thrust  On  the  Flanges 


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Comparative  Design  of  F/ange 
Ft  re  as 


Comparative  Flange  Freas  of 
O-tt,  /-H,  2-H,  & 3-H  Fashes 


Curves  Nos.  22-26 

Dl  -Dead Load,  LL  = Live  Load,  T=  Temperature 
PS = Rib  -Shortening  RN=  Rib-shortening 
neglected , R I = Rib-shortening  included. 


fvO/npa. 

Envelops 


•relive  /Reaction 
of  No-  end  One-h/’nged 
S/ee/  ftrches 

O-M  /St  SC) ’*  dtrzf-tdhtx  *ht*  - O 
/-A.  soco-ufA  * r-  ^-A$7  * robra-o 
ScbB^rOCbi  *S-BJJr  Sc  — O 

A—  - # - fen  it*  pj  * a-£  S-f 
S-  aFc  r / -dbcrdb’^F  c-saAb-7 

@ Reaction  locus  of  o-A 
® Reeclon  locus  of  /-A 
© Reeclon  Envelop  oF  o-A 

@ Reaction  Envelop  oF  t-A 
20/agram  20 


case  the  moments  and  thrust  produced  by  the  rib- 
shortening always  have  the  same  negative  sign  in 
the  case  of  a one-hinged  arch.  This  naturally  tends 
to  decrease  the  area  of  section  required,  if  the  area 
of  section  is  controlled  by  the  positive  moments  and 
thrust,  in  the  design,  as  is  usually  the  case  in  the 
one-hinged  arch. 

The  comparative  flange  areas  required  for  the 
three-,  two-,  one-,  and  no-hinged  arches  for  the  same 
form  of  the  arch-rib  under  the  same  specifications 
are  shown  in  the  following  table  and  on  curve  sheet 
25.  The  areas  given  for  the  three-hinged  arch  are 
exact,  while  those  for  the  two-,  one-,  and  no-hinged 
arches  are  obtained  from  a design  with  the  as- 


sumption 

/ 5 seed 

with  the 

effect  of  r 

ib-shortening 

included. 

Three- 

Two- 

One- 

No- 

Section. 

IIinged. 

Hinged. 

Hinged. 

Hinged. 

sq. in. 

sq.  in. 

sq.  in. 

sq.  in. 

0 

68.0 

68.4 

153.7 

119.5 

1 

73.0 

72.8 

127.0 

93.1 

2 

78.2 

78.1 

103.6 

75.9 

3 

82.4 

85.0 

84.0 

72.3 

4 

84.4 

90.7 

67.7 

65.0 

5 

84.1 

93.1 

67.7 

67.9 

6 

81.2 

94.9 

66.8 

72.8 

7 

77.7 

94.8 

66.0 

76.9 

8 

71.0 

92.3 

65.4 

78.4 

9 

65.0 

89.3 

64.9 

78.3 

10 

64.0 

89.0 

64.7 

77.3 

tions  are  the  largest  of  all.  The  two-hinged  arch 
requires  the  largest  area  in  the  sections  near  the 
middle  for  the  same  reason.  The  one-hinged  and 
no-hinged  arches  require  the  largest  areas  in  the 
sections  near  the  ends,  because  the  end  moment  is 
the  dominating  factor  in  those  sections.  In  com- 
paring the  areas  in  flanges  of  the  one-  and  no-hinged 
arches,  we  see  that  the  one-hinged  arch  requires 
comparatively  large  areas  in  ihe  sections  near  the 
ends  and  small  areas  in  the  sections  near  the  crown. 
The  areas  near  the  crown  of  the  one-hinged  arch 
are  rather  uniform,  because  the  thrusts  at  these 
points  are  large  and  about  equal  in  magnitude. 

The  comparative  theoretical  web  areas  required 
for  the  three-,  two-,  one-  and  no-hinged  arches  with 
the  same  form  of  the  rib  and  designed  under  the 
same  specifications  are  shown  in  the  following  table 


and  also 

in  curve 

sheet  26 : — 

Three- 

Two- 

One- 

No- 

Section. 

Hinged. 

Hinged. 

Hinged. 

Hinged 

1 

sq.  in. 

7.91 

sq. in. 

10.04 

sq.  in. 

19.59 

sq. in. 

22.83 

2 

6.95 

8.98 

16.76 

19.88 

3 

6.31 

8.20 

14.95 

18.22 

4 

5.77 

7.67 

13.03 

15.51 

5 

5.45 

7.45 

11.10 

13.58 

6 

6.10 

7.61 

9.55 

12.08 

7 

6.95 

7.92 

7.95 

10.87 

8 

7.61 

8.11 

6.89 

9.70 

9 

7.97 

8.12 

6.45 

9.18 

10 

8.13 

7.96 

5.75 

10.46 

14 


It  is  seen  that  the  no-hinged  arch  requires  the 
largest  web  area,  while  the  three-hinged  arch,  the 
least  of  all.  The  one-hinged  arch  requires  smaller 
web  areas  than  the  no-hinged  arch,  while  the  two- 
liinged  arch  requires  greater  areas  than  the  three- 
hinged  arch.  The  no-hinged  and  one-hinged  arches 
require  larger  web  areas  in  the  sections  near  the 
ends  and  smaller  areas  in  the  sections  near  the 
crown.  This  is  due  to  the  fact  that  the  shear  in 
the  no-hinged  and  one-hinged  arches  is  far  greater 
near  the  ends  than  in  the  sections  near  the  crown. 
On  the  other  hand,  the  web  areas  of  the  three-,  and 
two-hinged  arches  are  rather  uniform,  since  the 
shears  in  the  sections  of  the  three-  and  two-hinged 
arches  are  rather  uniform.  The  marked  difference 
is  caused  by  the  fixing  of  the  supports  in  the  one 
case,  and  the  hinging  of  the  supports  in  the  other. 

The  following  table  shows  the  results  of  the  de- 
sign based  upon  the  assumption  / 5 sec  0 


Section. 

Composition 

Moment  of 
Inertia. 

Ratio  of  I. 

See  0 . 

Proposed 
Assumption, 
of  I. 

0-2 

6z.s  6"  X 6”  X 9/16" 

3 Pis  14"  X 9/16" 

5 Pis  18"  X 7/8" 

1 Pis  18"  X 3/4" 

286,000 

2.49 

1.07 

250% 

2-4 

6z.s  6"  X 6"  X 9/16" 

3 Pis  14"  X 9/16" 

2 Pis  18"  X 3/4" 

1 Pis  18"  X 13/16" 

190,600 

1.59 

1.05 

160% 

4-6 

6z.s  6"  X 6"  X 9/16" 

3 Pis  14"  X 9/16" 

3 Pis  18"  X 5/16" 

123,800 

1.03 

1.03 

103% 

6-8 

6z.s  6"  X 6”  X 9/16" 

3 Pis  14"  X 9/16" 

1 Pis  18"  X 1/4" 

121,700 

1.02 

1.01 

101% 

8-10 

6z_s  6"  X6"  X 9/16" 

1 Pis  14"  X 9/16" 

1 Pis  18"  X 3/16" 

119,700 

1.00 

1.00 

100% 

It  is  seen  that  the  assumption  I ' sec  Q is  about 
right  for  the  sections  near  the  crown,  and  is  in  great 
error  for  the  sections  near  the  ends. 

DISCUSSION  ON  TRUE  DESIGN 
So  far  as  the  reactions  are  concerned,  the  effect 
of  final  design  as  compared  with  that  of  the  pre- 
liminary design,  assuming  I z sec.  e,  is  to  increase 
the  horizontal  reactions  when  the  load  is  near  the 
center  of  the  span  and  to  decrease  the  same  when 
the  load  is  near  the  ends.  There  is  no  appreciable 
change  in  the  vertical  reactions  for  the  two  de- 
signs. The  end  moments  have  been  affected  so 
greatly  that  under  the  full  loading,  a negative  end 


moment  is  produced  in  this  design,  which  was  not 
the  case  in  the  preliminary  design.  The  latter  has 
a serious  effect  in  designing  the  sections,  because 
the  sign  of  the  dead  load  moment  is  changed  to 
negative,  and  the  maximum  moments  at  various 
sections  are  thereby  changed. 

The  preliminary  design  by  assuming  I z sec.  o is 
far  from  being  correct.  It  makes  the  flange  areas 
at  the  various  sections  far  different  from  the  true 
values.  The  error  is  on  the  unsafe  side.  It  is 
necessary  that  a revised  design  be  made  in  de- 
signing a one-hinged  arch. 

The  preliminary  design  by  assuming  / ; sec.  e is 
nearest  to  the  true  value  for  the  two-hinged  arch, 
while  it  differs  most  in  the  one-hinged  arch.  The 
relative  error  of  the  preliminary  design  from  the 
true  design  of  the  no-,  one-,  and  two-hinged  arches 
with  same  dimensions  and  designed  to  carry  the 
same  load  is  shown  below : 


Section 

No-hinged 

One-hinged 

Two-liinged 

per  cent 

per  cent 

per  cent 

0 

—15.5 

—30.7 

5.2 

1 

—13.4 

—31.6 

—3.7 

2 

— 7.6 

—33.1 

—4.7 

3 

— 2.4 

—34.4 

—2.5 

4 

— 3.6 

—40.5 

2.4 

5 

8.7 

—33.3 

3.3 

6 

5.6 

—27.0 

7.6 

7 

4.1 

—21.0 

12.8 

8 

3.7 

—16.1 

12.1 

9 

2.5 

—12.8 

11.6 

10 

2.1 

—11.7 

13.7 

Thus,  we  see  that  the  error  in  the  one-hinged  arch 
is  the  largest  of  all.  This  is  due  to  the  presence  of 
the  large  moments  in  the  sections  near  the  ends  and 
the  large  thrust  in  the  sections  near  the  crown.  The 
variation  of  the  moments  and  thrusts  in  the  sections 
is  such  that  the  moment  of  inertia  in  the  various  sec- 
tions required  is  not  in  accordance  with  the  relation 
jz  sece. 

As  the  assumption  / s sec. o is  far  from  being  true, 
the  I-Curve  for  a one-hinged  arch  with  1/10  rise  is 
recommended  by  the  author,  by  means  of  which  a 
closer  result  can  be  secured  in  the  approximate  de- 
sign. A similar  curve  was  recommended  for  the  no- 
hinged  arch  by  P.  H.  Chen  in  his  thesis  above  named. 
The  comparative  values  are  shown  below : 


Section 

One-hinged 

N 

o-h 

inged 

0 

Io 

+ 

185% 

Io 

1 

~r 

120% 

1 

Io 

+ 

135% 

Io 

60% 

2 

Io 

+ 

100% 

Io 

_L 

30% 

3 

Io 

+ 

70% 

Io 

1 

~r 

15% 

4 

Io 

+ 

45% 

Io 

6 

Io 

+ 

15% 

Io 

+ 

10% 

8 

Io 

+ 

5% 

Io 

— 

10% 

10  In  I 


15 


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\ 

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-- 

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Fr&d/c 

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sec_ 

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. 

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-t- 

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o 

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5 . . V " ^ 

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t : -i 

’ fri-lrlrtirhiS 

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z.  i a 

f 

It  is  seen  from  the  curves  that  the  assumptions  of 
I s sec.®  and  1=  constant  are  nearly  correct  in  the 
sections  near  the  center,  while  they  are  too  small  in 
the  sections  near  the  ends.  The  exceedingly  large 
moments  near  the  ends  both  in  the  no-hinged  and 
one-hinged  arches  necessitate  the  use  of  the  large 
sections  and  accordingly  the  values  of  the  large 
moments  of  inertia. 

• 

DISCUSSION  OF  METHODS  OF  DEFLECTION 
COMPUTATIONS 

Deflections  in  the  one-hinged  arch-ribs  are  con- 
tributed by  two  factors,  that  due  to  thrust  and  that 
due  to  moment.  The  methods  used  by  the  author  in 
finding  these  deflections  are  rather  interesting  and 
are  here  given. 

(a)  DEFLECTION  DUE  TO  MOMENTS— The 
formulas  for  finding  the  horizontal  and  vertical  de- 
flections of  a curved  beam  were  given  in  equations 
(d)  and  (e)  respectively.  As  the  continuity  of  the 
one-hinged  arch  is  broken  at  the  crown  by  the 
presence  of  the  hinge,  the  formulas  could  not  be  ap- 
plied directly  to  the  entire  span.  A method  of 
separating  the  arch  into  two  cantilevers  is  intro- 
duced (Fig.  3).  For  example,  when  the  load  is  at 
a certain  point  on  the  left  half  of  the  span,  the  re- 
actions acting  on  the  left  cantilever  are  Hx  and  V2, 
while  those  on  the  right  cantilever  are  Hx  and  V2. 
lever  are  contributed  by  three  factors,  due  to  P,  Hx 
and  V2,  while  those  in  the  right  cantilever  are  con- 
tributed by  two  factors,  Hx  and  V2.  These  factors 
can  be  easily  found  by  considering  each  half  in 
turn  as  a cantilever  with  one  load  on  the  arch. 
Thus,  under  the  vertical  load  P on  the  left  cantilever, 
we  only  need  to  find  the  deflections  in  the  cantilever 
due  to  a horizontal  load  at  the  free  end,  a vertical 
load  at  the  free  end,  and  a vertical  applied  load. 
The  deflections  due  to  Hx  and  V2  in  the  left  half 
of  the  arch  are  just  the  same  in  magnitude  as  those 
due  to  Hx  and  V2  in  the  right  half  of  the  arch.  The 
final  value  of  the  vertical  or  horizontal  deflections 


can  be  easily  obtained  by  taking  the  algebraic  sum 
of  the  vertical  or  horizontal  deflections  contributed 
by  the  various  factors. 

If  the  deflection  diagrams  are  required  for  a unit 
load  at  different  positions  on  the  left  half  of  the 
span,  the  method  is  exceedingly  simple.  We  need 
only  to  draw  the  deflection  diagrams  of  the  canti- 
lever with  a unit  horizontal  load  at  the  free  end, 
and  a unit  vertical  load  at  different  positions  on  the 
cantilever  corresponding  to  the  positions  on  the 
arch.  These  are  obtained  as  unit  deflections.  The 
values  of  Hx  and  V2  are  then  found  for  different 
positions  of  the  unit  applied  load.  The  deflections 
due  to  II \ and  V 2 are  then  found  by  multiplying  the 
unit  deflections  by  the  values  of  Hx  and  V2.  Using 
proper  signs  of  deflections  contributed  by  each 
factor  and  taking  the  algebraic  sum,  the  deflections 
at  various  points  on  the  arch  can  be  found  for  dif- 
ferent positions  of  the  loading. 

The  method  has  several  advantages : first,  it  is 
simple  as  well  as  easy;  second,  it  offers  the  oppor- 
tunity of  studying  the  deflections  contributed  by 
each  factor;  third,  it  is  easy  to  discover  mistakes. 
By  means  of  the  graphical  method,  the  deflection  on 
the  cantilever  due  to  a unit  vertical  or  horizontal 
load  at  any  position  on  the  beam  can  be  easily  cal- 
culated. By  arranging  the  work  in  a systematic 
way,  errors  can  be  easily  discovered  by  comparison. 

The  method  is  applicable  in  any  of  the  four  cases : 
(1)  vertical  deflection  under  the  vertical  load,  (2) 
horizontal  deflection  under  the  vertical  load,  (3) 
vertical  deflection  under  the  horizontal  load,  (4) 
horizontal  deflection  under  the  horizontal  load.  The 
only  difference  lies  in  the  magnitude  and  direction 
of  II x and  V2  and  the  unit  horizontal  or  vertical  de- 
flections to  be  found  in  the  cantilevers. 

(b)  DEFLECTION  DUE  TO  THRUST— The 
general  formula  for  the  deflection  due  to  thrust  is 
given  by 

Deflection  = f*Tt 

in  which  T is  the  thrust  due  to  the  applied  load  P; 
while  small  t is  the  thrust  due  to  a load  unity  P" 
applied  at  the  point  whose  deflection  is  sought,  the 
direction  of  P"  being  the  same  as  the  direction  of 
of  the  deflection  required.  Let  H'  and  V'  be  the  hori- 
zontal and  vertical  shear  (not  in  the  normal  section) 
immediately  on  the  left  of  the  section  considered;  and 
H"  and  V",  those  due  to  P"  respectively.  Then,  by 
substitution,  the  general  formula  of  deflection  due  to 
thrust  becomes, 

Deflect /on  =J'bH’H"cos2e^  +J^‘VVr"sjn  *6^ 

which  is  applicable  to  the  four  cases  above  named,  that 
is,  the  vertical  and  horizontal  deflections  due  to  the 
horizontal  load. 


16 


p 


S3 


In  order  to  simplify  the  numerical  work(  the  above 
formula  may  be  transformed  into  a simpler  form  for 
each  of  the  different  cases.  For  example,  let  us  take 
the  ease  of  the  horizontal  deflection  under  the  vertical 
loading'.  Let  a unit  load  P'  be  applied  at  t and  the 
horizontal  deflection  at  5 be  found.  (Figs.  59  and 
60).  The  corresponding  points  on  the  other  half  of 
the  arch  are  called  s'  and  t'  In  examining  the  loading 
and  reactions  closely,  we  find  that  H'  and  H"  in  s-c 
and  s'-c  are  correspondingly  equal.  H'  in  a-s  is  also 
equal  to  H'  in  b-s'  under  the  load  P' ; while  under  the 
load  P" , IP  in  a-s  is  equal  to  H1  and  H’  in  b-s'  is  equal 
to  H.,.  Also  V"  in  t-c  and  t'-c  are  correspondingly 
equal.  V"  in  a-t  is  equal  to  V"  in  t'-b,  while  V"  in 
a-t=V1  and  V”  in  b-t  is  equal  to  V2.  Using  the  proper 
signs,  the  formula  is  given  by 

H'HcoseOj{jz  —J'  (Hr )H  coo  PQ Ylt 
+f(vi’V;')vs<n*Q$: -2fYVs,n*e& 

It  is  seen  that  the  terms  containing  Us  can  be 
neglected,  because  the  angle  of  is  not  greater 
than  30  degrees  and  the  squares  of  sines  will  be  small 
in  value.  The  net  result  of  the  horizontal  deflection 
due  to  Us  is  negligible  in  comparison  with  the  de- 
flections caused  by  Hs  in  the  thrusts  and  that  by  the 
moments.  By  means  of  transformations  the  fol- 
lowing formulas  are  derived  with  several  approxi- 
mations : 

For  vertical  deflection  due  to  thrust  under  the  verti- 
cal load ; 

D-ef'H'H'cos’e 


for  horizontal  deflection  due  to  thrust  under  the 
horizontal  load; 

D-zprH"C03‘e&  ~jJf(HrWl)coS‘e  j* 

-j'/q&H) 

and  for  vertical  deflection  due  to  thrust  under  the 
horizontal  load; 

D=2jCH'H"cos*e £§  -ffarH^Hcos^e ^ 

DISCUSSION  ON  DEFLECTIONS 

The  moments  and  thrusts  are  equally  important 
in  causing  the  deflections  in  the  one-hinged  arch. 
The  latter  is  especially  important  in  the  sections 
near  the  center.  In  considering  the  deflections 
caused  by  the  thrust,  the  vertical  forces  may  be 
neglected  without  appreciable  error.  Shear  can  be 
neglected  too. 

Under  the  vertical  loading  the  vertical  deflections 
near  the  end  of  the  arch  are  similar  to  those  of  the 
no-hinged  arch,  while  those  near  the  center  resemble 
in  the  form  those  of  the  three-hinged  arch.  As  a 
whole,  the  one-hinged  arch  is  subject  to  greater  ver- 
tical deflection  than  the  no-hinged  arch  and  less 
vertical  deflections  than  the  two-,  and  three-hinged 
arches  for  the  vertical  loading.  Thus,  the  no-hinged 
arch  is  the  most  favorable  in  stiffness  among  the 
four  types  of  arches,  so  far  as  vertical  deflections 
under  the  vertical  load  are  concerned.  The  one- 
hinged  arch  comes  the  next,  the  two-hinged  arch 
comes  third,  and  the  three-hinged  arch  is  the  most 
unfavorable  of  all.  Comparative  deflection  curves 
for  the  four  types  of  arches  with  the  same  dimen- 
sions and  designed  under  the  same  loading  are  pre- 
pared by  the  author  for  different  positions  of  the 
loading.  They  are  not  here  reproduced  due  to  the 
limit  of  space. 

The  same  relation  is  found  for  the  horizontal 
deflections  under  the  vertical  loading  for  the  four 
types  of  arches.  The  horizontal  deflections  of  the 
three-  and  two-hinged  arches  are  much  greater  than 
those  of  the  no-  and  one-hinged  arches,  while  the 
horizontal  deflections  of  the  no-  and  one-liinged 
arches  are  about  the  same.  Comparative  curves 
of  horizontal  deflections  under  the  vertical  load  for 
the  four  types  of  arches  are  also  prepared  by  the 
author. 

Under  the  horizontal  loading,  the  vertical  and 
horizontal  deflections  for  the  one-hinged  arch  are 
found  to  be  very  small.  This  seems  to  indicate  that 
the  vibrations  under  the  moving  train  are  low,  and 
the  effect  of  wind  is  also  rather  unimportant  for 
this  type  of  the  arch.  Comparative  curves  of  de- 
flections under  the  horizontal  loading  for  the  four 
types  of  the  arches  shows  that  the  one-hinged  arch 
has  greater  horizontal  and  vertical  deflections  than 
the  110-hinged  arch,  and  less  vertical  and  horizontal 


17 


deflections  than  the  two-  and  three-hinged  arches. 

The  comparative  maximum  and  minimum  deflec- 
tions for  the  four  types  of  arches  are  shown  in 
Fig.  70.  The  one-hinged  arch  is  again  seen  to  be 
stiffer  than  the  two-  and  three-hinged  arches,  and 
not  so  stiff  as  the  no-hinged  arch. 


The  yielding  of  a support  has  a serious  effect  on 
the  stresses.  With-  a horizontal  movement  of  the 
support  of  one  inch  away  from  the  original  position, 
an  additional  value  of  126.8  kips  is  increased  in  the 
horizontal  reactions.  This  increases  the  thrust  at 
the  crown  to  twice  the  original  value,  and  thus 


greatly  affects  the  safety  of  the  arch.  The  yielding 
of  supports  has  a similar  effect  on  the  two-  and  no- 
hinged  arches.  The  effect  on  the  one-hinged  arch 
is  a little  less  than  that  on  the  no-hinged  arch,  and 
much  greater  than  that  in  the  two-hinged  arch. 
For  the  same  amount  of  horizontal  movement,  the 
increase  in  the  horizontal  reaction  in  the  two-hinged 
arch  is  29.8  kips,  while  that  in  the  no-hinged  arch 
is  161.0  kips.  Thus,  we  see  that  a perfectly  firm 
foundation  is  required  for  these  three  types  of 
arches. 

Remarks 

The  main  feature  of  this  investigation  consists 
in  the  discovering  new  formulas  and  relations  for 
the  one-hinged  arch.  These  formulas  and  relations 
are  then  carefully  studied  with  the  aid  of  numerous 
curves  and  diagrams,  and  their  characteristics  are 
revealed  by  means  of  critical  comparisons.  Al- 
though the  main  purpose  of  this  thesis  consists  in 
searching  for  new  theoretical  facts,  it  is  hoped  that 
it  may  stimulate  investigations  in  a practical  treat- 
ment of  the  one-hinged  arch  with  regard  to  its  de- 
sign and  construction.  The  knowledge  of  the  sub- 
ject, however,  is  still  in  its  infancy.  Notwithstand- 
ing the  large  amount  of  study  and  the  labor  de- 
voted to  the  subject  treated  in  this  thesis,  other 
subjects  remain  to  be  investigated,  such  as  the 
economic  ratio  of  rise  to  span,  the  relative  merits 
of  solid  and  open  webs,  the  secondary  stresses,  etc. 


OF  ,U[;n,; 

m 2 „ m 


« »«r 


